If \(\xi _c\)  acts as the analog to \(T_c\)  in GL mean-field theory, this naturally raises the question, “Is the non-minimally coupled vortex a thermodynamic system at non-zero temperature?". The answer is clearly no. The Nielsen-Olesen vortex without gravity constitutes a static classical field configuration which is at zero temperature and has zero entropy. The zero temperature agrees with the fact that the fields have no average kinetic energy and the zero entropy is in accord with the fact we know everything about the field's configuration throughout spacetime; we are not ignorant of its configuration at any time and no information is hidden from us. The zero entropy is of course consistent with the zero temperature. When gravity is included, this can change only if the vortex acquires an event horizon. However, our gravitating vortex solutions are non-singular static solutions with no event horizon. The temperature and entropy are again zero and as before, the metric field, as well as the scalar and gauge field, are static throughout all of spacetime. In contrast, the BTZ black hole [1]}, [2]} has a non-zero temperature and entropy as it has an event horizon (for simplicity, assume no angular momentum or electric charge, only mass with a single horizon). Note that the BTZ spacetime has a timelike Killing vector outside the event horizon but like the Schwarzschild black hole in \(3+1\)  dimensions, it has no timelike Killing vector inside the event horizon [3]}. This implies that there is no coordinate transformation that can put the metric in static form inside the event horizon so that an outside observer is ignorant of the metric configuration inside at any particular time. Simply put, information is hidden from us behind the event horizon [4]}. Note that in contrast, our non-singular gravitating static vortex has a timelike Killing vector throughout spacetime and no information is hidden from us (see also [5]}, [6]}, [7]} for a related discussion).
